I can't yet pull off the notable farming feats described by great players ... Making 200-300k per hour farming Sky
Rule number one, don't always believe the numbers people post about EXP/Hr, gil/Hr, DMG/WS, etc... It's like real life fishermen, where the fish they caught tends to grow a few inches each time they tell the story...
Starting with Alchemy (perhaps a poor choice), I've leveled my skill up to the mid 20s. So far I'm not seeing any good profits even when farming materials (and even then, I could have made more money farming something else with the time I invested).
I understand that you need higher level skill to HQ on items as this helps with making profits. I understand that there's a lot more to making money off of crafting than just plugging the materials in and getting gold to come out. But after spending a couple of hours studying the markets on my server, checking where to get materials the cheapest, and calculating profit margins, the outlook does not look too great.
Well, you seem to have understood the basics of crafting and crafting for gil, which means you have to potential of becoming a great craftsman, eventually.
However, given the amount of players that have leveled crafting to some degree at this point in time, the main problem with crafting for gil is the competition. If there is something simple most every player can craft for profit, you can expect a lot of players to be doing it, thus producing a huge amount of the given result, driving down the prices. Hence, small profit margins or even no profit at all.
The key to crafting for gil is to find the niches in the market for which other crafters have trouble meeting the requirements.
You already gave the most obvious example of such a 'niche': high level crafting. The higher your level, the greater your chance of HQing, thus increasing your profit margin. E.g. for your lvl 20ish Alchemy, you will find it hard (if at all possible) to make a profit turning beehive chips into beeswax, because most other Alchemists are at Tier 3 (level 55+), outproducing you because they will consistently HQ the wax. The sheer amount of high level crafters has, at this point in time, resulted in NQ products (gear mostly) being sold at loss, because crafters are trying to make that gear+1 item to hit the big profit. Unless you can compete with them, stay out of this market segment.
A second way to explore niches is focusing on very specific requirements for certain synths, i.e. main craft/sub craft/key item combinations. A simple example for your Alchemist could be the silver bullet recipe, which requires you to also have Goldsmithing leveled to some degree, or the several elemental anima recipes which require the key item 'Anima synthesis'. A notorious example from the past would be the Noble's Bed for one of the Moogle Quests requiring Woodworking (78), Goldsmithing (50) and Clothcraft (56). The stranger (read 'more uncommon') the combination, the higher the chance you will only find a few crafters capable of doing that specific synth, hence lower competition and higher possible profit.
Making gil also means being able to 'read' the market. Many crafters make the mistake of jumping on what seems a promising product, only to flood the market and to realize half the server's population of crafters had about the same idea. Diversification is the key to making profit.
Finally, making gil relies heavily on profit margins. Evidently, your profit is determined by a lot of factors, amongst the most important are the total price of your ingredients, the amount of work needed to finish one unit of product, the selling rate of your product, the selling price of your product, the amount of gil lost on breaks and the probability (risk) of breaking, the required level of product (NQ, HQ1, ...) and finally the taxes you need to pay.
Some of these factors cannot be influenced by you. The selling price, for instance, is something you have little to no influence on. One small exception to this rule is products sold to NPC vendors: not all NPC vendors are alike! In some cases guild vendors offer more for a specific product than regular vendors. Vendors of your own nation might offer more too and in general you will get more from NPC vendors if you have high fame. All in all, these differences are small or neglectable (except the difference between guild vendors and regular vendors which can be up to 10% or more).
The selling rate, however, in some cases can be controlled. The most obvious example, again, is selling to NPC. Especially with regular vendors, who normally accept unlimited amounts of your product, the selling rate is only limited by your production rate. Most of the 'interesting' recipes for profit crafting to NPC stuff have been nerved by SE though (rusty caps or coffee muffins anyone?). What is left usually yields very low profit. Your Alchemist could probably make a few K/Hr selling holy water to NPC, if you can HQ the recipe.
Something you have more control over is the cost of your ingredients. Buying smart is the key here. If you can make a lot of gil (you define what 'a lot' is) farming the ingredients, by all means do so. Often however, it will be more efficient to just buy the ingredients and take a smaller profit margin. Often also, it might be more advisable to farm something different to sell and cover the cost of your ingredients over directly farming the ingredients. E.g. don't go farming sheep skins if you need 50 of them to make leather (which would take you hours probably), just farm a few K worth of other stuff to sell and use that money to buy the skins from the guild vendor (if you are lucky they are sold for about 70 gil a piece).
Reducing the cost of ingredients might also require crafting them yourself, perhaps even on another mule character, specialized in that particular ingredient. E.g. your Alchemist (at higher levels) is perfect for making the elemental anima your Bonecrafter or Goldsmith needs to make Reraise items. Crafting them might allow you to lower the price of the ingredients significantly (compared to buying from AH). Keep in mind though, that if you craft base ingredients yourself, you are also taking the risk of breaking. So, if you are only gaining a small amount crafting your own base ingredients, and a single break could potentially wipe out the profit you make synthing several stacks of product (e.g. water anima), consider buying instead of crafting yourself.
Buying smart, finally, also means picking the right time to buy base ingredients and stock up on them. Don't buy ingredients if the price is too high, unless the profit margin on your final product justifies the extra cost. Wait for the ingredient cost to go down again before buying. This also means you could (should) be buying ingredients you intend to use for profit crafting later on when their price has fallen dramatically and store them for future use. Of course, only buy them when you are fully confident the price will go up again eventually (which requires you again to be able to 'read' the market).
My question is this: am I making a huge mistake? Is it even possible to profit from crafting anymore, or am I just wasting my time and money?
Unless you intend to become really, really good at crafting and understanding your server's market, don't expect to get filthy rich overnight from crafting.
It is generally not that hard to find a way to make profit from crafting, though, the question is, are you prepared to put in the effort for the profit you can expect out of it. The less effort you put into it, the lower your profit will be. If you are happy to craft for hours in a row, making 20K/Hr, I'm sure there are plenty of recipes to be found allowing you just that. On the other hand, if you are only willing to spend a few hours a week on crafting to make up for the 300K or more of expenses you have on average for your job leveling, ... good luck finding that niche.
Personally, I have leveled multiple crafts on my main and on several dedicated mules to as high levels as I could get them just for fun, because I generally like crafting. I never really crafted for money, although I found it a challenge to find a skill up path that would actually allow me to level my crafts at profit or at least break even. However, I now have the possibility to craft just about all the stuff my main needs (food, gear, furniture, ammo, weapons, ...) and every time I craft something myself, I am making a profit (because otherwise I would have needed to buy the same thing at AH prices). I can't put an exact number on that hidden form of profit, but over the 6 years I've been playing this game, I suspect it to be quite important.